More notes on the Calculus Blue Multivariable Volume 1 videos on YouTube by “Prof Ghrist Math”.<\/p>\n
Chapter 2 introduces curves in the plane and surfaces in 3-space with implicit and parametric definitions for curves in the plane and for surfaces in 3-space. \u00a0They also introduce the names and images for all of the quadratic surfaces (so we’ve left the linear world in this chapter). \u00a0This is a foreshadowing of the process of the course (start with the linear and then move to the nonlinear).<\/p>\n
01.02.00 (0:35) “Curves and surfaces: Intro”<\/p>\n
01.02.01 (5:10) “Implicit and parametric curves and surfaces”. \u00a0Two ways to define curves. \u00a0Implicitly or parametrically. \u00a0Implicit: “the solutions to an equation yields a curve in the plane”. \u00a0Parametric: “specifying coordinates as a function of a parameter”. \u00a0We’ll want to move between these representations. \u00a0Surfaces in 3d can also be expressed implicitly or parametrically (requires 2 parameters).<\/p>\n
Examples: parameterization for a surface expressed implicitly (he shows the conversion where you set x = s, y = t, and then express z); go from a parameterization to an implicit equation (the example has a square root so he suggests caution).<\/p>\n
01.02.02 (0:29) “Some examples please…?”. \u00a0it’s important to learn the quadratic surfaces.<\/p>\n
01.02.03 (5:33) “Examples of quadratic surfaces”. \u00a0start with the sphere, then modify to get the ellipsoids, change a sign to get a hyperboloid, with two negatives signs get a 2-sheeted hyperboloid, then an elliptic paraboloid, hyperbolic paraboloids, cones (degenerate hyperbola), cylinders.<\/p>\n
Then the narrator provides a reassurance: this isn’t about memorizing these or drawing pictures of these; it’s just worth recognizing them. \u00a0Then the narrator mentions these won’t come back for a while and names surface integrals as an application. \u00a0These are in there as a motivation to build up geometry and algebra skills.<\/p>\n
01.02 (0:26) “The big picture”. \u00a0Previous chapter was lines and planes. \u00a0This one was curves and surfaces. \u00a0Progression from linear to nonlinear is what will happen in the course.<\/p>\n","protected":false},"excerpt":{"rendered":"
More notes on the Calculus Blue Multivariable Volume 1 videos on YouTube by “Prof Ghrist Math”. Chapter 2 introduces curves in the plane and surfaces in 3-space with implicit and parametric definitions for curves in the plane and for surfaces in 3-space. \u00a0They also introduce the names and images for all of the quadratic surfaces […]<\/p>\n","protected":false},"author":8032,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_feature_clip_id":0,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2},"jetpack_post_was_ever_published":false},"categories":[1010,157885],"tags":[],"class_list":["post-188","post","type-post","status-publish","format-standard","hentry","category-math","category-multivariable"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/p7E5LF-32","jetpack-related-posts":[{"id":186,"url":"https:\/\/archive.blogs.harvard.edu\/siams\/2019\/07\/22\/notes-on-calculus-blue-volume-1-chapter-1\/","url_meta":{"origin":188,"position":0},"title":"Notes on “Calculus Blue” Volume 1, Chapter 1","author":"siams","date":"22 July 2019","format":false,"excerpt":"These notes are on the Calculus Blue videos by Ghrist on YouTube. \u00a0He emphasizes that the math will involve substantial (and worthwhile) work, which I really appreciate. 01 (0:51) \"Vectors & matrices: Intro\" \u00a0\"Your journey is not a short one\". \u00a0To learn \"calculus, the mathematics of the nonlinear\", prepare with\u2026","rel":"","context":"In "Math"","block_context":{"text":"Math","link":"https:\/\/archive.blogs.harvard.edu\/siams\/category\/math\/"},"img":{"alt_text":"","src":"","width":0,"height":0},"classes":[]},{"id":83,"url":"https:\/\/archive.blogs.harvard.edu\/siams\/2018\/03\/03\/the-vector-calculus-bridge-project\/","url_meta":{"origin":188,"position":1},"title":"The vector calculus bridge project","author":"siams","date":"3 March 2018","format":false,"excerpt":"Reading about \"The vector calculus bridge project\" (Tevian Dray and Corinne Manogue at Oregon State). http:\/\/math.oregonstate.edu\/bridge\/talks\/OSU.pdf http:\/\/physics.oregonstate.edu\/~tevian\/bridge\/papers\/FEdgap.pdf Their takeaways: * key calculus idea: the differential (not limits) * key derivative idea: rates of change (not slopes) * key integral idea: total amounts (not areas\/volumes) * key curves\/surfaces idea: \"use what\u2026","rel":"","context":"In "Math"","block_context":{"text":"Math","link":"https:\/\/archive.blogs.harvard.edu\/siams\/category\/math\/"},"img":{"alt_text":"","src":"","width":0,"height":0},"classes":[]},{"id":58,"url":"https:\/\/archive.blogs.harvard.edu\/siams\/2017\/08\/25\/vector-calculus-earlier-in-the-semester\/","url_meta":{"origin":188,"position":2},"title":"Vector calculus earlier in the semester?","author":"siams","date":"25 August 2017","format":false,"excerpt":"Flux is a particularly central scientific and mathematically idea that appears in the context of a multivariable calculus course. \u00a0Given a velocity vector field and a surface, the flux of the vector field through the surface tells us the rate at which fluid is flowing through the surface. \u00a0This leads\u2026","rel":"","context":"In "Math"","block_context":{"text":"Math","link":"https:\/\/archive.blogs.harvard.edu\/siams\/category\/math\/"},"img":{"alt_text":"","src":"","width":0,"height":0},"classes":[]},{"id":193,"url":"https:\/\/archive.blogs.harvard.edu\/siams\/2019\/07\/22\/notes-on-calculus-blue-volume-1-chapter-3-and-4\/","url_meta":{"origin":188,"position":3},"title":"Notes on Calculus Blue Volume 1, Chapters 3, 4, 5, 6","author":"siams","date":"22 July 2019","format":false,"excerpt":"More of Calculus Blue by Prof Ghrist Math. Chapter 3 is on coordinates and distance (see section 12.1 of the 6th edition of Hughes-Hallett). 01.03 (0:36) \"Coordinates: intro\". \u00a0Review coordinates and see it in data. 01.03.01 (2:16) \"coordinates & many dimensions\". \u00a0from curves and surfaces we'll head on. \u00a0plane, then\u2026","rel":"","context":"In "Math"","block_context":{"text":"Math","link":"https:\/\/archive.blogs.harvard.edu\/siams\/category\/math\/"},"img":{"alt_text":"","src":"","width":0,"height":0},"classes":[]},{"id":141,"url":"https:\/\/archive.blogs.harvard.edu\/siams\/2019\/06\/19\/hughes-hallett-et-al-chapter-11-differential-equations\/","url_meta":{"origin":188,"position":4},"title":"Hughes-Hallett et al Chapter 11: Differential equations","author":"siams","date":"19 June 2019","format":false,"excerpt":"11.1: What is a differential equation? Starts with an example: what sets the rate at which a person learns a new task? \u00a0Defines a diff eq and a solution to a diff eq. Defines order of a diff eq. \u00a0Example 1 is showing a function is not a solution to\u2026","rel":"","context":"In "Math"","block_context":{"text":"Math","link":"https:\/\/archive.blogs.harvard.edu\/siams\/category\/math\/"},"img":{"alt_text":"","src":"","width":0,"height":0},"classes":[]},{"id":134,"url":"https:\/\/archive.blogs.harvard.edu\/siams\/2019\/06\/17\/hughes-hallett-et-al-chapter-8-using-the-definite-integral\/","url_meta":{"origin":188,"position":5},"title":"Hughes-Hallett et al Chapter 8: Using the definite integral","author":"siams","date":"17 June 2019","format":false,"excerpt":"For the course \"Integrating and Approximating\" our focus will be on multivariate integration, vector calculus, and differential equations. \u00a0In the past, I've used a number of texts for Multivariable, but appreciate the four-fold perspective (tables, graphs, formulas, words) that is used in Hughes-Hallett et al. A few chapters of single\u2026","rel":"","context":"In "Math"","block_context":{"text":"Math","link":"https:\/\/archive.blogs.harvard.edu\/siams\/category\/math\/"},"img":{"alt_text":"","src":"","width":0,"height":0},"classes":[]}],"_links":{"self":[{"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/posts\/188","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/users\/8032"}],"replies":[{"embeddable":true,"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/comments?post=188"}],"version-history":[{"count":1,"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/posts\/188\/revisions"}],"predecessor-version":[{"id":189,"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/posts\/188\/revisions\/189"}],"wp:attachment":[{"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/media?parent=188"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/categories?post=188"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/tags?post=188"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}